Computer-Readable Recording Medium Storing Quantum Computation Support Program, Quantum Computation Support Method, and Information Processing Apparatus

This application is based upon and claims the benefit of priority of the prior Japanese Patent Application No. 2023-172746, filed on Oct. 4, 2023, the entire contents of which are incorporated herein by reference.

The embodiments discussed herein are related to a computer-readable recording medium storing a quantum computation support program, a quantum computation support method, and an information processing apparatus.

By using quantum chemical computation, it is possible to examine characteristics of an unknown substance. Highly accurate energy of a substance is computed by the quantum chemical computation. Energy in a ground state of a quantum multi-body system may be computed by, for example, a variational quantum eigensolver (VQE). The VQE is an algorithm using a noisy intermediate-scale quantum computer (NISQ) without error correction. The VQE is expected to be applied to drug discovery and new substance discovery.

“SPSA (Simultaneous Perturbation Stochastic Approximation)—A Method for System Optimization” [online], The Johns Hopkins University Applied Physics Laboratory, [searched on Sep. 19, 2023], Internet <URL: https (scheme name)://www.jhuapl.edu/SPSA/(host name+path name)>, and Knizia, Gerald, and Garnet Kin-Lic Chan, “Density Matrix Embedding: A Simple Alternative to Dynamical Mean-Field Theory”, Physical review letters, Nov. 2, 2012, Vol. 109, Iss. 18 are disclosed as related art.

According to an aspect of the embodiments, a non-transitory computer-readable recording medium stores a quantum computation support program for causing a computer to execute a process including: training a regression model in which parameter values are explanatory variables and computation results of quantum computation are objective variables, based on a correspondence relationship between the computation results for each of a plurality of times of quantum computation for each of the parameter values according to a quantum circuit that includes parameters and the parameter values set in the quantum computation; and specifying a solution of the quantum computation by using the regression model.

The object and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the claims.

It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention.

In the VQE, parameters are optimized based on results of evaluating quantum circuits in the NISQ. For the optimization of the parameters, for example, a method such as a simultaneous perturbation stochastic approximation (SPSA) is used. The VQE may be used as a subroutine in a density matrix embedding theory (DMET). The DMET is a quantum embedding theory for computing a quantity that does not depend on a frequency such as ground state characteristics of an infinite system.

A solution obtained by search for an optimization solution using a quantum device including noise such as a NISQ may deviate from a true optimal solution due to an influence of noise. Such a solution error due to the influence of noise is a problem that occurs not only in the VQE but also in many quantum computations using the NISQ, such as a quantum approximate optimization algorithm (QAOA) which is an algorithm for combinatorial optimization and quantum computation using the VQE as a subroutine like the DMET.

In one aspect, an object of the present disclosure is to reduce an influence of noise in quantum computation.

Hereinafter, the present embodiments will be described with reference to the drawings. Each embodiment may be implemented by combining a plurality of embodiments within a range without contradiction.

A first embodiment is a quantum computation support method for suppressing an influence of noise in quantum computation.

1 FIG. 1 FIG. 10 10 is a diagram illustrating an example of a quantum computation support method according to the first embodiment.illustrates an information processing apparatusfor performing the quantum computation support method. For example, by executing a quantum computation support program, the information processing apparatusis able to perform the quantum computation support method.

10 11 12 11 10 12 10 The information processing apparatusincludes a storage unitand a processing unit. The storage unitis, for example, a memory or a storage device included in the information processing apparatus. The processing unitis, for example, a processor or an arithmetic circuit included in the information processing apparatus.

11 2 2 2 2 The storage unitstores a quantum circuitincluding a parameter. For example, the quantum circuitincludes a quantum gate that indicates a gate operation according to the parameter. As the quantum circuitincluding a parameter, for example, there is a quantum circuit for obtaining a ground state of a quantum multi-body system. For example, in a case where a ground state of a quantum multi-body system is obtained by a VQE, the quantum circuitincluding a quantum gate that performs a gate operation of rotation designated by an angle parameter is used.

2 12 7 12 3 Based on a computation result of the quantum computation using the quantum circuit, the processing unitcomputes a solutionwith a reduced influence of noise. For example, the processing unitfirst acquires history informationindicating a correspondence relationship between a computation result and a parameter value.

12 2 12 1 2 1 For example, the processing unitrepeatedly executes processing (optimization) of updating the parameter value in a direction in which the computation result approaches a ground state, and processing of calculating the computation result based on the gate operation according to the quantum circuitin which the updated parameter value is set. For example, the processing unitmay cause a quantum computerto execute the gate operation according to the quantum circuitin which the updated parameter value is set, and may calculate the computation result based on a measurement result by the quantum computer.

1 2 1 3 4 6 6 In a case where the quantum computeris a NISQ, noise is included in the gate operation and the measurement according to the quantum circuitby the quantum computer. For this reason, when points indicating a relationship between the parameter values and the computation results indicated in the history informationare plotted in a graph, variations due to noise occur at the plotted points. In the optimization processing, for example, a parameter value at a point where a value of a computation result indicating energy of a ground state is smallest is an optimization solution. However, an accuracy of the computation result may be lowered due to the influence of noise, and there is no guarantee that the optimization solutionis a true solution indicating the ground state.

12 5 2 5 Accordingly, the processing unittrains (performs training of) a regression modelbased on a correspondence relationship between computation results for each of a plurality of times of computation (quantum computation) for each of the parameter values according to the quantum circuitand the parameter values set in the quantum computation. The regression modelis a model in which parameter values are explanatory variables and computation results are objective variables.

12 5 5 For example, the processing unittrains the regression modelby using a Gaussian process (GP) or a kernel method. As a method of generating the regression model, there is a method such as linear regression or polynomial regression using a feature vector, but the linear regression or the polynomial regression may be regarded as a special case of the Gaussian process or the kernel method.

12 7 5 12 5 7 The processing unitspecifies the solutionof the quantum computation by using the regression model. For example, the processing unitspecifies a value of an explanatory variable at which a value of an objective variable takes an extreme value (for example, a minimum value) in the regression model, as the solutionof the quantum computation.

5 3 7 5 7 2 7 As described above, by training the regression modelbased on the history informationand specifying the solutionby using the regression model, it is possible to obtain the solutionwith a reduced influence of noise included at the time of computation based on the quantum circuit. For example, it is possible to obtain the solutionhaving a small error from a true optimal solution.

3 12 3 7 Since the influence of noise is allowed to be included in the history information, the processing unitmay acquire the history informationby using an optimization method with a small processing load such as an SPSA. Accordingly, the processing load for specifying the solutionwith a reduced influence of noise is small.

12 5 12 5 12 7 7 The processing unitmay specify a solution in consideration of uncertainty of estimation of the value of the objective variable in the regression model. For example, the processing unitcomputes uncertainty of estimation of an objective variable according to an explanatory variable of the regression modelby the Gaussian process or the kernel method. The processing unitspecifies the solutionof the quantum computation from a range of values of the explanatory variable in which the uncertainty of estimation of the objective variable is equal to or greater than a predetermined value (reliability of an estimated value is high). Accordingly, it is possible to obtain the solutionhaving a small error from a true optimal solution.

5 The Gaussian process or the kernel method is effective for learning a smooth function from a relatively small amount of data. In the current NISQ, a number of iterations of optimization is not very large, and there is not a large amount of data. For this reason, by using the Gaussian process or the kernel method, it is possible to generate the regression modelwith a high accuracy even when the number of iterations of optimization is small.

12 12 Based on the correspondence relationship between the computation results and the parameter values, the processing unitmay train regression model candidates to which each of a plurality of regularization parameters is applied. In this case, the processing unitdetermines a regression model based on a predetermined condition from among the regression model candidates for each of the plurality of regularization parameters.

2 2 2 2 12 5 For example, a computation result using the quantum circuithaving parameters is a smooth function with respect to the parameters when there is no noise. In a case of the simple quantum circuithaving one parameter, the function is a trigonometric function. More generally, even in a case of the quantum circuithaving a plurality of parameters, when one parameter is focused on, a function corresponding to the representative quantum circuitis a trigonometric function. Accordingly, the processing unitmay determine the regression modelbased on consistency with the trigonometric function from among the regression model candidates for each of the plurality of regularization parameters.

5 5 5 5 By determining a regression model based on the predetermined condition from among the regression model candidates for each of the plurality of regularization parameters, it is possible to obtain the regression modelusing an appropriate regularization parameter even when the appropriate regularization parameter is unknown. When there is no influence of noise in the optimization, it may be known that the regression modelis represented by a trigonometric function. In such a case, by specifying a regression model having high consistency with the trigonometric function among the plurality of regression model candidates as the regression model, a regression model candidate to which an appropriate regularization parameter is applied may be correctly specified as the regression model.

A second embodiment is to suppress an influence of noise in quantum chemical computation using a VQE and improve an accuracy of a solution. The quantum chemical computation is to obtain energy of a quantum multi-body system (a type of molecule+a ground function). The ground function is a function that is a source constituting a molecular orbital. By obtaining the energy of the quantum multi-body system, it is possible to grasp characteristics of the molecule. The energy to be obtained includes ground energy and excitation energy. The ground energy is energy in a state where a molecular structure is stable. The excitation energy is energy in a state where a molecular structure is unstable. Among possible states of the molecule, energy in a state where the energy is minimum is the ground energy.

2 FIG. 300 400 20 400 300 is a diagram illustrating an example of a system configuration according to the second embodiment. In the second embodiment, a quantum computer systemand a terminal deviceare coupled to each other via a network. The terminal devicetransmits a request for quantum computation to the quantum computer systemin response to an operation from a user.

300 100 200 100 200 100 200 200 100 The quantum computer systemincludes a classical computerand a quantum computer. The classical computerand the quantum computerare coupled to each other via a communication interface. The classical computeris a Neumann-type computer, and performs processing such as generation of a quantum circuit and optimization of parameters used for computation of the quantum circuit. The quantum computeris a computer that performs the quantum chemical computation by performing an operation based on a quantum gate on a quantum bit. The quantum computerperforms quantum chemical computation by a VQE algorithm in accordance with the quantum circuit and the parameters generated by the classical computer.

3 FIG. 100 101 102 101 109 101 101 101 is a diagram illustrating an example of hardware of devices included in the quantum computation system. The entirety apparatus of the classical computeris controlled by a processor. A memoryand a plurality of peripheral devices are coupled to the processorvia a bus. The processormay be a multiprocessor. The processoris, for example, a central processing unit (CPU), a microprocessor unit (MPU), or a digital signal processor (DSP). At least a part of the function realized by the processorexecuting a program may be realized by an electronic circuit such as an application-specific integrated circuit (ASIC) or a programmable logic device (PLD).

102 100 102 101 102 101 102 The memoryis used as a main storage device of the classical computer. The memorytemporarily stores at least a part of an operating system (OS) program or an application program to be executed by the processor. The memorystores various types of data to be used for the processing by the processor. As the memory, for example, a volatile semiconductor storage device such as a random-access memory (RAM) is used.

109 103 104 105 106 107 108 108 a b. The peripheral devices coupled to the businclude a storage device, a graphics processing unit (GPU), an input interface, an optical drive device, a device coupling interface, and network interfacesand

103 103 100 103 103 The storage devicewrites and reads data electrically or magnetically to and from a built-in recording medium. The storage deviceis used as an auxiliary storage device of the classical computer. The storage devicestores an OS program, an application program, and various types of data. As the storage device, for example, a hard disk drive (HDD) or a solid-state drive (SSD) may be used.

104 104 21 104 104 21 101 21 The GPUis a calculation device that performs image processing. The GPUis an example of a graphic controller. A monitoris coupled to the GPU. The GPUdisplays an image on a screen of the monitorin accordance with a command from the processor. As the monitor, a display device using organic electro luminescence (EL), a liquid crystal display device, or the like is used.

22 23 105 105 22 23 101 23 A keyboardand a mouseare coupled to the input interface. The input interfacetransmits signals transmitted from the keyboardor the mouseto the processor. The mouseis an example of a pointing device, and other pointing devices may be used. Examples of the other pointing devices include a touch panel, a tablet, a touch pad, a track ball, or the like.

106 24 24 24 24 The optical drive devicereads data recorded in an optical discor writes data to the optical discby using laser light or the like. The optical discis a portable-type recording medium in which data is recorded such that the data is readable by reflection of light. Examples of the optical discinclude a Digital Versatile Disc (DVD), a DVD-RAM, a compact disc read-only memory (CD-ROM), a CD-recordable (CD-R), a CD-rewritable (CD-RW), and the like.

107 100 25 26 107 25 107 26 27 27 27 The device coupling interfaceis a communication interface for coupling the peripheral devices to the classical computer. For example, a memory deviceor a memory reader and writermay be coupled to the device coupling interface. The memory deviceis a recording medium provided with a function of communicating with the device coupling interface. The memory reader and writeris a device that writes data to a memory cardor reads data from the memory card. The memory cardis a card-type recording medium.

108 20 108 20 108 108 a a a a The network interfaceis coupled to the network. The network interfacetransmits and receives data to and from another computer or a communication device via the network. The network interfaceis, for example, a wired communication interface that is coupled to a wired communication device such as a switch or a router by a cable. The network interfacemay be a wireless communication interface that is coupled, by radio waves, to and communicates with a wireless communication device such as a base station or an access point.

108 200 101 200 108 200 101 108 b b b. The network interfaceis an interface for coupling to the quantum computer. The processortransmits a quantum circuit to the quantum computervia the network interfaceand causes the quantum computerto execute quantum computation. The processoracquires a result of the quantum computation via the network interface

100 10 100 3 FIG. With the hardware as described above, the classical computermay realize processing functions of the second embodiment. The information processing apparatusdescribed in the first embodiment may also be realized by substantially the same hardware as that of the classical computerillustrated in.

100 100 100 103 101 103 102 100 24 25 27 103 101 101 For example, the classical computerrealizes the processing functions of the second embodiment by executing a program recorded in a computer-readable recording medium. The program in which a content of processing to be executed by the classical computeris described may be recorded in any of various recording media. For example, the program to be executed by the classical computermay be stored in the storage device. The processorloads at least a part of the program in the storage deviceto the memory, and executes the program. The program to be executed by the classical computermay be recorded in a portable-type recording medium such as the optical disc, the memory device, or the memory card. The program stored in the portable-type recording medium may be executed after the program is installed in the storage deviceunder the control of the processor, for example. The processormay read the program directly from the portable-type recording medium and execute the program.

200 210 220 210 220 220 The quantum computerhas a control deviceand a quantum device. The control deviceexecutes a gate operation on a quantum bit in the quantum device according to the quantum circuit. The quantum devicehas a plurality of quantum bits. The quantum deviceis, for example, a quantum processing unit (QPU).

300 100 200 200 In the quantum computer system, by the classical computerand the quantum computeroperating in cooperation with each other, quantum chemical computation by a VQE is performed. At this time, since the quantum computeris a NISQ, an accuracy of an optimization solution obtained by the VQE is reduced due to an influence of noise.

4 FIG. 200 200 1 2 N is a diagram illustrating an example of the quantum chemical computation by the VQE. In the quantum chemical computation by the VQE, the quantum computerperforms quantum measurement based on an initial value of a parameter θ (a set of variables of a circuit corresponding to electron excitation). For example, the quantum computercalculates a values of each of a plurality of divided Hamiltonians (H, H, . . . , and H) in accordance with the quantum circuit (N is a natural number).

30 30 31 32 33 34 35 36 37 39 An Ansatz circuit is included in a quantum circuitused for the quantum measurement. The quantum circuitis a quantum computation model described by combining a plurality of quantum gates. The quantum gates include an Hadamard gate, an X gate, a Y gate, a Z gate, an S gate, a T gate, and a rotation gateto a rotation gate, and so on.

31 A gate operation of the Hadamard gateis represented by Expression (1).

32 A gate operation of the X gateis represented by Expression (2).

33 A gate operation of the Y gateis represented by Expression (3).

34 A gate operation of the Z gateis represented by Expression (4).

35 A gate operation of the S gateis represented by Expression (5).

36 A gate operation of the T gateis represented by Expression (6).

37 A gate operation of the rotation gatethat rotates a state around an X axis by an angle indicated by the parameter θ is represented by Expression (7).

38 A gate operation of the rotation gatethat rotates a state around a Y axis by the angle indicated by the parameter θ is represented by Expression (8).

39 A gate operation of the rotation gatethat rotates a state around a Z axis by the angle indicated by the parameter θ is represented by Expression (9).

4 FIG. 30 Although all of the quantum gates illustrated inare 1-qubit gates, quantum gates (for example, 2-qubit gates) that operate a plurality of quantum bits are also used in the quantum circuit.

30 37 39 30 200 30 As the quantum gates used in the quantum circuit, there are quantum gates for which a rotation angle is designated by the parameter θ, like the rotation gatesto. When the quantum circuitincludes a plurality of rotation gates, the value of the parameter θ for each rotation gate is designated. The quantum computerexecutes quantum computation by applying the designated value of the parameter θ and performing a gate operation according to the quantum circuiton the quantum bit. As a result, a plurality of divided Hamiltonians are calculated.

100 100 100 200 The plurality of calculated Hamiltonians are added by the classical computerto obtain an expected value of energy of the entire system. The classical computerperforms optimization of one or a plurality of parameters θ based on the expected value of energy. For example, the classical computerupdates the parameter θ in a direction in which the expected value of energy decreases. After the parameter θ is updated, the quantum computercomputes the expected value of energy based on the updated parameter θ again.

Such computation of the expected value of energy and the update of the parameter are repeated until ground energy is obtained. An example of the parameter optimization method is an SPSA. The SPSA is an algorithm in consideration of inclusion of noise in function evaluation. Although it is indicated that a good solution is obtained by taking an average for a number of optimization trials in the SPSA, a returned solution is easily influenced by noise in one trial with randomness. Under the influence of noise, an error occurs between a solution output by the optimization and a true optimal solution.

Accordingly, to reduce the influence of noise, estimation using a Gaussian process or a kernel method may be considered. At this time, when training based on the Gaussian process or the kernel method is performed in the middle of the optimization, estimation computation by the Gaussian process or the kernel method is performed a plurality of times, and a computation amount becomes excessive. A combination method of the estimation by the Gaussian process or the kernel method and the optimization method such as the SPSA for which results in the NISQ are known is not obvious, and there is a possibility that the influence of noise may not be sufficiently suppressed.

As methods of optimizing while removing the influence of noise, there are an implicit filtering algorithm (IMFIL), Bayesian optimization, and the like. The IMFIL is a method of removing high-frequency vibration from a function by discrete Fourier transform, and does not remove noise with randomness. For example, the IMFIL is a method of optimizing while estimating a smoothed function by using the discrete Fourier transform. Because the IMFIL is an optimization method, there is a possibility that noise that may not be reduced by the discrete Fourier transform may be reduced by post-processing using a regression model described later.

The Bayesian optimization is an optimization method for estimating a model of a function by using a Gaussian process or a kernel method. Because the Bayesian optimization performs estimation in each iteration of optimization, a computation amount is larger than that of other methods such as a stochastic gradient descent method.

There is also a technology in which noise reduction by a Gaussian process or a kernel method is used for optimizing in the NISQ. For example, “GP+IMFIL” uses a Gaussian process or a kernel method for an initial point of the IMFIL (local search method). Although a better initial point may be obtained, an influence of noise may remain in a solution returned by the local search method.

300 Accordingly, the quantum computer systemestimates a true optimal solution by the Gaussian process or the kernel method by using a history of optimization performed in the process of the VQE computation.

5 FIG. 40 40 40 is a diagram illustrating an example of estimation by a Gaussian process or a kernel method using an optimization history. A graphindicates a relationship between a value of a parameter (one dimension) included in an energy computation expression and an energy value. A horizontal axis of the graphindicates a parameter value, and a vertical axis of the graphindicates an energy value.

40 41 40 Points in the graphindicate parameter values obtained by optimization processing performed in the process of the VQE computation and energy values obtained from computation results of the quantum circuit when these parameter values are input. A pointat which an energy value is minimum is a solution of the quantum computation by the VQE computation. As illustrated in the graph, the energy value obtained by the quantum computation for each parameter has an error due to an influence of noise.

42 42 42 42 It is possible to generate a regression modelindicating the relationship between the parameter values and the energy values based on the history of optimization performed in the process of the VQE computation. The regression modelis a function in which the parameter values are explanatory variables and the energy values are objective variables. The regression modelis obtained as a result of estimation by a Gaussian process or a kernel method, for example. A curve coupling expected values of the energy values according to the parameter values is the regression model.

43 42 41 43 A parameter value at a pointat which an energy value is minimum in the regression modelis shifted from a parameter value at the pointat which the energy is minimum in the quantum computation. By setting the parameter value and the energy value corresponding to the pointas a solution for the optimization computation, a solution in which the influence of noise is suppressed is obtained.

6 FIG. 100 110 120 130 is a block diagram illustrating an example of functions of the quantum computer system. The classical computerincludes a quantum computation management unit, a parameter update unit, and a regression model training unit.

110 110 400 110 200 The quantum computation management unitacquires a quantum circuit for computing energy of a quantum multi-body system such as a molecule. For example, the quantum computation management unitacquires a quantum circuit for performing optimization computation by the VQE algorithm from the terminal device. The quantum computation management unitinstructs the quantum computerto perform quantum computation based on the acquired quantum circuit.

110 110 200 110 120 110 200 For example, the quantum computation management unitsets initial values for a plurality of parameter values before the first quantum computation. The quantum computation management unitacquires a computation result based on the quantum circuit parameterized by the plurality of parameters from the quantum computer. After that, the quantum computation management unitacquires updated parameter values from the parameter update unit. The quantum computation management unitinstructs the quantum computerto perform computation based on the quantum circuit in which the updated parameter values are set.

200 110 110 110 110 120 Based on the measurement result by the quantum computer, the quantum computation management unitcomputes an energy value according to the set parameter value. For example, the quantum computation management unitsets a sum of values of divided Hamiltonians as energy. When the energy value converges, the quantum computation management unitsets the energy value at that time as ground energy. When the energy value has not converged, the quantum computation management unitinstructs the parameter update unitto update the parameters.

130 110 200 200 110 In a case where an optimal solution of the parameter in a case where the influence of noise is reduced is acquired from the regression model training unit, the quantum computation management unitinstructs the quantum computerto perform quantum computation based on this optimal solution. Based on the measurement result by the quantum computer, the quantum computation management unitcomputes an energy value according to the optimal solution.

120 120 110 For each quantum computation, the parameter update unitupdates all or some values of the plurality of parameters in a direction in which the energy value decreases. The parameter update unitnotifies the quantum computation management unitof the updated plurality of parameter values.

130 130 130 130 110 Based on the plurality of parameter values for each quantum computation and the energy values obtained by the quantum computation, the regression model training unittrains a regression model indicating a relationship between the parameter values and the energy values. For example, the regression model training unitgenerates a regression model by a Gaussian process or a kernel method. Based on the regression model obtained by the training, the regression model training unitcomputes an optimal solution for the parameter values with a reduced influence of noise. The regression model training unitinstructs the quantum computation management unitto perform quantum computation based on the optimal solution of the parameter values.

100 6 FIG. For example, the function of each element in the classical computerillustrated inmay be realized by causing a computer to execute a program module corresponding to the element.

300 By such a quantum computer system, ground energy computation with a reduced influence of noise is performed.

7 FIG. 300 10 20 is a diagram illustrating an example of ground energy computation processing. The ground energy computation processing by the quantum computer systemis roughly divided into optimization processing (step S) by the VQE and expected value measurement processing (step S) by the optimal solution with a reduced influence of noise.

11 10 110 200 200 First, quantum state preparation processing (step S) is performed in the optimization processing (step S). For example, the quantum computation management unitinstructs the quantum computerto perform a gate operation for setting a quantum bit to a predetermined initial state. The quantum computersets the quantum bit to the initial state as instructed.

11 12 110 200 200 200 200 110 After the quantum state preparation processing (step S) is completed, expected value measurement processing (step S) is performed. For example, the quantum computation management unitinstructs the quantum computerto execute a quantum gate operation based on a quantum circuit for performing a VQE. According to the instruction, the quantum computerperforms the quantum gate operation. After the quantum gate operation according to the quantum circuit, the quantum computermeasures a state of the designated quantum bit. For example, the quantum computerrepeats the quantum gate operation based on the quantum circuit and the measurement a predetermined number of times, and transmits an expected value of the state of the quantum bit to be measured to the quantum computation management unit.

110 120 13 13 120 120 The quantum computation management unitcomputes energy based on the expected value of the state of the quantum bit. The parameter update unitperforms parameter update processing (step S). In the parameter update processing (step S), the parameter update unitupdates values of at least some of the plurality of parameters so as to reduce energy. For example, the parameter update unitupdates the parameter values by an SPSA.

11 12 13 11 12 13 After the parameter values are updated, the quantum state preparation processing (step S), the expected value measurement processing (step S), and the parameter update processing (step S) are performed based on the quantum circuit parameterized by the updated parameters. The quantum state preparation processing (step S), the expected value measurement processing (step S), and the parameter update processing (step S) are repeated until the computed energy value is determined as ground energy.

10 102 103 20 By the optimization processing (step S), sets of parameter values and energy values when these parameter values are applied are recorded as history information in the memoryor the storage device. By acquiring the recorded history information, the expected value measurement processing (step S) by the optimal solution with a reduced influence of noise is executed.

21 22 23 20 Regression model training processing (step S), quantum state preparation processing (step S), and expected value measurement processing (step S) are executed in the expected value measurement processing (step S) by the optimal solution with a reduced influence of noise.

21 130 130 In the regression model training processing (step S), for example, the regression model training unittrains a regression model by a Gaussian process or a kernel method. Based on the trained regression model, the regression model training unitcomputes the optimal solution with a reduced influence of noise. The optimal solution is a parameter value that minimizes an energy value.

22 110 200 200 After the optimal solution is obtained, the quantum state preparation processing (step S) is performed based on the optimal solution. For example, the quantum computation management unitinstructs the quantum computerto perform a gate operation for setting a quantum bit to a predetermined initial state. The quantum computersets the quantum bit to the initial state as instructed.

22 23 110 200 200 200 200 110 110 After the quantum state preparation processing (step S) is completed, the expected value measurement processing (step S) is performed. For example, the quantum computation management unitinstructs the quantum computerto execute a quantum gate operation based on a quantum circuit for performing a VQE. According to the instruction, the quantum computerperforms the quantum gate operation. After the quantum gate operation according to the quantum circuit, the quantum computermeasures a state of the designated quantum bit. For example, the quantum computerrepeats the quantum gate operation based on the quantum circuit and the measurement a predetermined number of times, and transmits an expected value of the state of the quantum bit to be measured to the quantum computation management unit. Based on the expected value of the state of the quantum bit, the quantum computation management unitcomputes energy according to the optimal solution.

20 22 23 100 Energy computation may be performed by a quantum simulator or the like by using the optimal solution with a reduced influence of noise in the expected value measurement processing (step S) by the optimal solution with a reduced influence of noise. In this case, quantum simulation by quantum simulator software is performed instead of the quantum state preparation processing (step S) and the expected value measurement processing (step S). The quantum simulation is processing of obtaining a change in a state of a quantum bit in response to the quantum gate operation indicated by the quantum circuit, by computation using the classical computer. By performing the energy computation with the quantum simulator, an energy value from which the influence of noise is removed is obtained.

21 The regression model training processing (step S) based on the optimization history information will be described in detail next.

8 FIG. 8 FIG. is a flowchart illustrating an example of a procedure of the regression model training processing. Hereinafter, the processing illustrated inwill be described in order of step numbers.

101 130 [Step S] The regression model training unitacquires optimization history information. The optimization history information is a history of sets of parameters and observed values (for example, energy values).

n n N d d It is assumed that a parameter of the quantum circuit is x. xis a vector of R(Ris a real number space in which a number of dimensions is d (a natural number)). n={1, 2, . . . , and N} (N is a number of iterations of optimization processing). xis a solution obtained in the optimization processing.

n n 1 1 2 2 N N 200 It is assumed that a measurement result when the quantum circuit having xas the parameter is executed by the quantum computeris y. At this time, history information is {(x, y), (x, y), . . . , and (x, y)}.

102 130 [Step S] Based on the acquired history information, the regression model training unittrains a regression model by a Gaussian process or a kernel method. The regression model is represented by a function in which parameters are explanatory variables and observed values are objective variables.

103 130 130 N [Step S] The regression model training unitspecifies a parameter value that minimizes the function estimated by the Gaussian process or the kernel method, as an optimal solution with a reduced influence of noise. For example, in the function obtained by the Gaussian process or the kernel method, the regression model training unitspecifies x* that minimizes an estimated value of the function in the vicinity of x, as the optimal solution with a reduced influence of noise.

130 130 The regression model training unitmay specify the optimal solution with a reduced influence of noise, from among the parameter values with which reliability of the energy value estimated by the regression model is equal to or greater than a certain value. For example, in a case where a number of samples (sets of parameter values and energy values) of the history information is small, the reliability of estimation by the Gaussian process or the kernel method is low. In this case, the regression model training unitspecifies the optimal solution with a reduced influence of noise from among the parameter values with which the reliability is equal to or greater than a certain level, thereby suppressing an incorrect optimal solution from being specified due to the small number of samples.

104 130 [Step S] The regression model training unitoutputs the optimal solution with a reduced influence of noise.

130 As described above, by training the regression model based on the optimization history information, it is possible to obtain the optimal solution with a reduced influence of noise. By using the Gaussian process or the kernel method as the regression model, the regression model training unitmay determine the optimal solution in consideration of uncertainty of estimation.

Details of the Gaussian process or the kernel method will be described below.

In the Gaussian process or the kernel method, it is assumed that X is a parameter space, and “k: X×X→R” (R is a real number space) is a kernel. The kernel represents similarity between two parameters, and as the kernel, for example, a Matern kernel, a Squared Exponential kernel, or the like is used.

The history information that is sets of parameters and

1 1 2 2 N N n i j i n measurement values for respective expected value measurements in the optimization processing is {(x, y), (x, y), . . . , and (x, y)}. It is assumed that Kis an n-th order square matrix in which (i, j) component is k(x, x) (i and j are natural numbers). It is assumed that a column vector whose i component is k(x, x) when x is a parameter is k(x).

n It is assumed that λ is a regularization parameter (λ>0). The regularization parameter is a value by which a regularization term included in an expression representing a regression model is multiplied. The regularization term is a term for smoothing noise. By using the regularization parameter, how much noise is smoothed is adjusted. An estimation μ(x) in the Gaussian process or the kernel method is given by Expression (10) below.

n Uncertainty σ(x) of estimation is given by Expression (11) below.

n n n The uncertainty of estimation indicates a level of possibility that a value of the estimation μ(x) is correct. A width of a confidence interval of the estimation μ(x) is defined by the uncertainty of estimation. As the width of the confidence interval is narrower, the probability that the value of the estimation μ(x) is correct is higher (reliability is higher).

When such a Gaussian process or a kernel method is applied to the history information of the VQE, the parameter θ of an angle in the VQE is used as a parameter x in Expression (10) and Expression (11). A value obtained by the estimation is an energy value. By estimating the energy value according to the parameter by a Gaussian process or a kernel method, uncertainty of this energy value may also be calculated.

9 FIG. 50 50 50 51 51 52 52 n n n n is a diagram illustrating an example of the regression model based on the Gaussian process or the kernel method. A horizontal axis of a graphindicates a parameter value and a vertical axis of the graphindicates an energy value. Points in the graphindicate samples included in the history information. Based on the parameter values and the energy values indicated by the samples, the Gaussian process or the kernel method is learned, and a regression modelis obtained. The regression modelis a function represented by Expression (10). A width of a confidence intervalin an energy value direction is represented by Expression (11). For example, the confidence intervalis represented by [μ(x)−σ(x), μ(x)+σ(x)].

10 FIG. n n n n 50 130 is a diagram illustrating an example of a solution based on the regression model by the Gaussian process or the kernel method. For example, a constraint condition “σ(x)≤c” (c is a positive real number) is provided in a case where an optimal solution with a reduced influence of noise is specified from among parameter values with which reliability is equal to or greater than a certain level. In the example of the graph, a region that satisfies the constraint condition is a region surrounded by broken lines. Outside the region surrounded by the broken lines, the reliability of the estimation μ(x) is low (uncertainty σ(x) is large). For this reason, the regression model training unitcomputes optimization “min μ(x)” only in the region surrounded by the broken lines.

130 53 53 n n As described above, in a case where the number of samples is small, the regression model training unitdoes not search the estimation μ(x) from the entire region, but obtains a pointthat minimizes μ(x) by narrowing down to a region having low uncertainty, and sets a parameter value of the pointas an optimal solution x* with a reduced influence of noise. This may be expressed as Expression below.

Accordingly, the optimal solution is specified in an interval with high reliability. Consequently, it is possible to obtain a highly reliable optimal solution.

130 130 n n n n At the time of minimizing the function, the regression model training unitmay set a parameter x having small uncertainty σ(x) and small estimation μ(x) as the optimal solution. For example, the regression model training unitsets a parameter x that minimizes “μ(x)+a×σ(x)” (a is a real number larger than 0) as the optimal solution.

As described above, it is possible to obtain the optimal solution with a reduced influence of noise by training the regression model using the Gaussian process or the kernel method based on the optimization history information and setting the parameter value that minimizes the energy value in this regression model as the optimal solution.

Since training based on the Gaussian process or the kernel method is performed as post-processing of the optimization method, the computation amount may be reduced as compared with a method in which training based on the Gaussian process or the kernel method is performed a plurality of times during optimization. Since it is post-processing of the optimization method, it may be used in combination with arbitrary optimization method.

By specifying an optimal solution in a region with high reliability, an optimal solution with high reliability is obtained.

A third embodiment is to, when an appropriate regularization parameter is unknown, allow an optimal solution to be calculated by computation using the appropriate regularization parameter.

The regularization parameter is a parameter indicating how much noise is smoothed when the Gaussian process or the kernel method is learned. Because an intensity of noise is unknown in the NISQ, it is difficult to know in advance how to set the regularization parameter.

300 0 0 0 Many of functions when energy is minimized by using a quantum circuit such as a VQE are trigonometric functions when one dimension of a parameter is focused. Accordingly, in the quantum computer system, the regularization parameter is adjusted so as to well represent the trigonometric function (to increase consistency). Assuming that a parameter to be optimized is x, a trigonometric function representing energy is, for example, “a sin (x+x)+b” or “a sin (2 (x+x))+b” (a, b, and xare real numbers).

11 FIG. 11 FIG. is a flowchart illustrating an example of a procedure of the regression model training processing including the adjustment of the regularization parameter. Hereinafter, the processing illustrated inwill be described in order of step numbers.

201 130 [Step S] The regression model training unitacquires optimization history information. The optimization history information is a history of sets of parameters and observed values (for example, energy values).

n n N d d It is assumed that a parameter of the quantum circuit is x. xis a vector of R(Ris a real number space in which a number of dimensions is d (a natural number)). n={1, 2, . . . , and N} (N is a number of iterations of optimization processing). xis a solution obtained in the optimization processing.

n n 1 1 2 2 N N 200 It is assumed that a measurement result when the quantum circuit having xas the parameter is executed by the quantum computeris y. At this time, history information is {x, y), (x, y), . . . , and (x, y)}.

202 130 130 −4 −3 −2 [Step S] The regression model training unitacquires a list L of regularization parameters. For example, the regression model training unitreads a list L prepared in advance from a memory. The list L is, for example, [10, 10, and 10].

203 130 [Step S] From the list L, the regression model training unitselects one of unselected regularization parameters. It is assumed that the selected regularization parameter is λ (λ∈L).

204 130 [Step S] By applying the selected regularization parameter λ, the regression model training unittrains a regression model (regression model candidate) by a Gaussian process or a kernel method by using the history information.

205 130 130 1 2 r r+1 r+2 r+s 1 2 r r+1 r+2 r+s d [Step S] The regression model training unitgenerates sample data for trigonometric function learning. For example, the regression model training unitgenerates “z, z, . . . , and z; z, z, . . . , and z∈R” as the sample data. Among them, “z, z, . . . , and z” is used to generate a model represented by a trigonometric function. “z, z, . . . , and z” is used to compute an error of the generated model.

130 N N N For example, the regression model training unitgenerates the sample data by changing arbitrary one component of x(d-dimensional vector), which is an optimization solution, to a neighboring value. For example, an i-th component (1≤i≤d) of the sample data is the neighboring value of the i-th component of x, which is the optimization solution. A j-th component (1≤j≤d, j≠i) other than the i-th component of the sample data has the same value as the j-th component of z.

206 130 1 2 r r+1 r+2 r+s 1 2 r r+1 r+2 r+s [Step S] Based on the regression model obtained by the Gaussian process or the kernel method, the regression model training unitcomputes an estimated value of each sample data. Each estimated value of “z, z, . . . , and z; z, z, . . . , and z” is referred to as “y′, y′, . . . , and y′; y′, y′, . . . , and y”.

207 130 130 130 1 1 2 2 r r 0 0 k k [Step S] The regression model training unitsets, as training data, sets “(z, y′), (z, y′), . . . , and (z, y′)” of r pieces of sample data and estimated values. By using the training data, the regression model training unitlearns a trigonometric function “f (x)=a sin (x+x)+b)”. For example, the regression model training unitlearns values of constants “a, b, and x” such that a difference between f (x) and y′ when “x=z” (1≤k≤r) is set decreases.

208 130 130 130 130 130 r+1 r+2 r+s r+1 r+2 r+s r+1 r+2 r+s r+1 r+2 r+s m m [Step S] The regression model training unitcomputes an error of the learned trigonometric function. For example, the regression model training unitobtains a value of the trigonometric function by using s pieces of sample data “z, z, . . . , and z” as explanatory variables of the trigonometric function. The regression model training unitsets values of the obtained trigonometric function as “y′, y″, . . . , and y″”. The regression model training unitcomputes an error between “y′, y′, . . . and y′” and “y″, y″, . . . , and y″”. For example, the regression model training unitsets a maximum value of “y′−y″″ (r+1≤m≤r+s)” as the error. The error is represented by Expression (13) below.

A regression model candidate having a smaller error may be determined to have higher consistency with the trigonometric function.

209 130 130 203 130 210 [Step S] The regression model training unitdetermines whether there is an unselected regularization parameter. When there is an unselected regularization parameter, the regression model training unitcauses the processing to proceed to step S. When all the regularization parameters have been selected, the regression model training unitcauses the processing to proceed to step S.

210 130 130 [Step S] The regression model training unitspecifies a regularization parameter having a minimum error as an optimal regularization parameter. The regression model training unitspecifies the regression model candidate generated by using the specified regularization parameter, as a regression model for estimating an optimal solution with a reduced influence of noise.

211 130 130 N [Step S] The regression model training unitspecifies, as the optimal solution, a parameter value that minimizes energy in a regression model corresponding to the specified regularization parameter. For example, the regression model training unitspecifies, as the optimal solution x*, a parameter value that minimizes an estimated value of the function in the vicinity of xin the function defining the regression model.

212 130 [Step S] The regression model training unitoutputs the optimal solution.

As described above, an appropriate regularization parameter is specified, and an optimal solution based on a regression model generated by using history information of optimization processing using the regularization parameter is output. Accordingly, even when the appropriate regularization parameter is unknown, it is possible to obtain an optimal solution using the appropriate regularization parameter.

Although optimization processing by the VQE is performed in the second and third embodiments, the method for specifying an optimal solution described in the second and third embodiments may be applied to computation results using a method other than the VQE.

The embodiments are exemplified above, the configuration of each unit described in the embodiments may be replaced with another unit having the same function. Arbitrary other components or processes may be added. Arbitrary two or more configurations (features) of the embodiments described above may be combined.

All examples and conditional language provided herein are intended for the pedagogical purposes of aiding the reader in understanding the invention and the concepts contributed by the inventor to further the art, and are not to be construed as limitations to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although one or more embodiments of the present invention have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.